Unordered pair: Difference between revisions
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In [[mathematics]], an '''unordered pair''' is a [[Set (mathematics)|set]] that has exactly two elements. In contrast, an [[ordered pair]] consists of a first element and a second element, which need not be distinct. An unordered pair may also be referred to as a '''two-element set''', a '''pair set''', or (rarely) a '''binary set'''. |
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{{Unreferenced|date=November 2009}} |
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A '''binary set''' is a [[Set (mathematics)|set]] with (exactly) two distinct elements, or, equivalently, a set whose [[cardinality]] is [[two]]. |
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A common notation for unordered pairs is {''a'',''b''}, where ''a'' and ''b'' are the elements of the set. However, if ''a'' = ''b'', then {''a'',''b''} = {''a''}, and the set is not an unordered pair but a [[singleton (mathematics)|singleton]]. |
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Examples: |
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* The set {''a'',''b''} is binary. |
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* The set {''a'',''a''} is not binary, since it is the same set as {''a''}, and is thus a [[singleton (mathematics)|singleton]]. |
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An unordered pair is a [[finite set]]; its [[cardinality]] (number of elements) is 2. |
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In [[axiomatic set theory]], the existence of binary sets is a consequence of the [[axiom of empty set]] and the [[axiom of pairing]]. From the axiom of empty set it is known that the set <math>\emptyset = \{\}</math> exists. From the axiom of pairing it is then known that the set |
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<math>\{\emptyset,\emptyset\} = \{\emptyset\}</math> exists, and thus the set <math>\{\{\emptyset\},\emptyset\}</math> exists. This latter set has two elements. |
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In [[axiomatic set theory]], the existence of unordered pairs is required by an axiom, the [[axiom of pairing]]. |
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==See also== |
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* [[ordered pair]] |
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* [[binary relation]] |
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{{DEFAULTSORT:Binary Set}} |
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[[Category:Basic concepts in set theory]] |
[[Category:Basic concepts in set theory]] |
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[[de:Paarmenge]] |
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[[eo:Duera aro]] |
[[eo:Duera aro]] |
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[[zh:二元集合]] |
[[zh:二元集合]] |
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Revision as of 15:55, 26 March 2010
In mathematics, an unordered pair is a set that has exactly two elements. In contrast, an ordered pair consists of a first element and a second element, which need not be distinct. An unordered pair may also be referred to as a two-element set, a pair set, or (rarely) a binary set.
A common notation for unordered pairs is {a,b}, where a and b are the elements of the set. However, if a = b, then {a,b} = {a}, and the set is not an unordered pair but a singleton.
An unordered pair is a finite set; its cardinality (number of elements) is 2.
In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.